Nonlinear Stochastic Operator Equations (eBook)

George Adomian (Autor)

eBook Download: PDF

2014 | 1. Auflage
304 Seiten
Elsevier Science (Verlag)
978-1-4832-5909-3 (ISBN)

Infos

Nonlinear Stochastic Operator Equations deals with realistic solutions of the nonlinear stochastic equations arising from the modeling of frontier problems in many fields of science. This book also discusses a wide class of equations to provide modeling of problems concerning physics, engineering, operations research, systems analysis, biology, medicine. This text discusses operator equations and the decomposition method. This book also explains the limitations, restrictions and assumptions made in differential equations involving stochastic process coefficients (the stochastic operator case), which yield results very different from the needs of the actual physical problem. Real-world application of mathematics to actual physical problems, requires making a reasonable model that is both realistic and solvable. The decomposition approach or model is an approximation method to solve a wide range of problems. This book explains an inherent feature of real systems-known as nonlinear behavior-that occurs frequently in nuclear reactors, in physiological systems, or in cellular growth. This text also discusses stochastic operator equations with linear boundary conditions. This book is intended for students with a mathematics background, particularly senior undergraduate and graduate students of advanced mathematics, of the physical or engineering sciences.

Inhalt

Front Cover 1
Nonlinear Stochastic Operator Equations 4
Copyright Page 5
Table of Contents 8
Foreword 12
Preface 14
Acknowledgments 16
Chapter 1. Introduction 18
References 19
Chapter 2. Operator Equations and the Decomposition Method 20
2.1. Modeling, Approximation, and Reality 20
2.2. The Operator Equations 24
2.3. The Decomposition Method 27
2.4. Evaluation of the Inverse Operator L-1 and the y0 Term of the Decomposition for Initial or Boundary 
28 
References 34
Suggested Further Reading 34
Chapter 3. Expansion of Nonlinear Terms: The An 
36 
3.1. Introduction 36
3.2. Calculation of the An Polynomials for Simple Nonlinear Operators 37
3.3. The An Polynomials for Differential Nonlinear Operators 41
3.4. Convenient Computational Forms for the An Polynomials 43
3.5. Linear Limit 46
3.6. Calculation of the An Polynomials for Composite Nonlinearities 46
References 50
Suggested Further Reading 50
Chapter 4. Solution of Differential Equations 51
4.1. General Method and Examples 51
4.2. Calculating a Simple Green's Function 55
4.3. Green's Function by Decomposition 57
4.4. Approximating Difficult Green's Functions 59
4.5. Polynomial Nonlinearities 64
4.6. Negative Power Nonlinearities 71
4.7. Decimal Power Nonlinearities 74
4.8. Product Nonlinearities 76
4.9. Anharmonic Oscillator Systems 84
4.10. Limiting Case: The Harmonic Oscillator 87
4.11. Extensions to Stochastic Oscillators 88
4.12. Asymptotic Solutions 89
4.13. The Question of Stability 104
References 104
Chapter 5. Coupled Nonlinear Stochastic Differential Equations 105
5.1. Deterministic Coupled Differential Equations 105
5.2. Stochastic Coupled Equations 110
5.3. Generalization to n Coupled Stochastic Differential Equations 110
Suggested Further Reading 114
Chapter 6. Delay Equations 115
6.1. Definitions 115
6.2. Solution of Delay Operator Equations 117
Reference 122
Suggested Further Reading 123
Chapter 7. Discretization versus Decomposition 124
7.1. Discretization 124
7.2. A Differential-Difference Equation 130
7.3. Difference Equations and the Decomposition Method 131
7.4. Some Remarks on Supercomputers 131
References 133
Suggested Further Reading 133
Chapter 8. Random Eigenvalue Equations 134
References 139
Suggested Further Reading 139
Chapter 9. Partial Differential Equations 140
9.1. Solving m-Dimensional 
140 
9.2. Four-Dimensional Linear Partial Differential Equation 146
9.3. Nonlinear Partial Differential Equation 148
9.4. Some General Remarks 150
9.5. The Heat Equation 152
9.6. Inhomogeneous Heat Equation 155
9.7. Asymptotic Decomposition for Partial Differential Equations 156
Note 157
Reference 157
Suggested Further Reading 158
Chapter 10. Algebraic Equations 159
Part I: Polynomials 159
10.1. Quadratic Equations by Decomposition 159
10.2. Cubic Equations 172
10.3. Higher-Degree Polynomial Equations 175
10.4. Equation with Negative Power Nonlinearities 180
10.5. Equations with Noninteger Powers 182
10.6. Equations with Decimal Powers 184
10.7. Random Algebraic Equations 186
10.8. General Remarks 187
Part II: Transcendental Equations 187
10.9. Trigonometric Equations 187
10.10. Exponential Cases 191
10.11. Logarithmic Equation: Purely Nonlinear Equations 195
10.12. Products of Nonlinear Functions 198
10.13. Hyperbolic Sine Nonlinearity 203
10.14. Composite Nonlinearities 204
Part III: Inversion of Matrices 214
10.15. Discussion: Inversion by Decomposition 214
10.16. Convergence 220
10.17. Decomposition into Diagonal Matrices 220
10.18. Systems of Matrix Equations 231
10.19. Inversion of Random Matrices 233
10.20. Inversion of Very Large Matrices 234
References 239
Suggested Further Reading 239
Chapter 11. Convergence 240
11.1. The Convergence Question for the Nonlinear Case 240
11.2. Estimating the Radius of Convergence 244
11.3. On the Calculus 246
11.4. Some Remarks on Convergence 247
References 247
Chapter 12. Boundary Conditions 248
12.1. Linear Boundary Conditions 248
12.2. Treatment of Inhomogeneous Boundary Conditions 249
12.3. General Boundary Operators and Matrix Equations 251
12.4. Random Boundary Operators 252
12.5. Random Inhomogeneous Boundary Conditions 253
12.6. Linear Differential Equations with Linear Boundary Conditions 254
12.7. Deterministic Operator Equations with Random Input and Random Boundary Conditions 262
12.8. Stochastic Operator Equations with Linear Boundary Conditions 265
12.9. Linear Stochastic Equations with Nonlinear Boundary Conditions 266
12.10. Linear Differential Equations with Nonlinear Boundary Conditions 269
12.11. Coupled Nonlinear Stochastic Differential Equations 275
12.12. Coupled Linear Deterministic Equations with Coupled Boundary Conditions 277
12.13. Coupled Equations and Coupled Boundary Conditions 283
12.14. Boundary Conditions for Partial Differential Equations 284
12.15. Summary 286
Reference 291
Suggested Further Reading 291
Chapter 13. Intergral and Integro-Differential Operators 292
Chapter 14. On Systems of Nonlinear Partial Differential Equations 296
References 299
Chapter 15. Postlude 300
References 301
Index 302
Errata for "Stochastic Systems" 304

Produkt-Details

• Erscheint lt. Verlag: 9.5.2014
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Informatik ► Theorie / Studium
• Themenwelt: Naturwissenschaften ► Physik / Astronomie
• Themenwelt: Technik
• ISBN-10: 1-4832-5909-9 / 1483259099
• ISBN-13: 978-1-4832-5909-3 / 9781483259093